Name: spring-mass-damper laplace block diagram
Uploaded: 2024-11-21T14:54:07.000Z
Duration: 1 min 20 s
Channel: JoVE

Considere a equação diferencial de segunda ordem de um sistema mola-massa-amortecedor. O sistema é transformado no domínio de Laplace sob condições iniciais zero.
A equação é então reorganizada para isolar a saída, que pode ser interpretada como sinais que entram em blocos com funções de transferência específicas.
A saída é obtida integrando duas vezes ou pós-multiplicando de acordo.
Para simplificar, os sinais do lado direito são conectados, levando à representação final do diagrama de blocos do sistema.
Uma simplificação adicional pode ser alcançada fatorando o termo do ciclo de feedback interno, resultando em um diagrama de blocos alternativo.
O modelo de diagrama de blocos também pode incorporar variáveis internas que representam aceleração e velocidade.
Como 1/s corresponde à integração no domínio de Laplace, a aceleração é inicialmente integrada para obter a velocidade e, posteriormente, a velocidade é integrada para produzir o sinal de deslocamento.
A função de transferência do sistema é encontrada movendo o bloco nos sinais de entrada e feedback para o lado direito do comparador e simplificando o loop de feedback interno. A equação resultante é a função de transferência do sistema.

relação entre equações matemáticas e diagramas de blocos

Relation between Mathematical Equations and Block Diagrams

Em um sistema mola-massa-amortecedor, a equação diferencial de segunda ordem descreve o comportamento dinâmico do sistema. Quando transformada no domínio de Laplace sob condições iniciais zero, essa equação pode ser efetivamente analisada e manipulada. A transformação no domínio de Laplace converte equações diferenciais em equações algébricas, simplificando o processo de isolamento da saída.

<img alt="PT_Equation1" src="/files/ftp_upload/16105/16105_PT_Equation_001.svg" style="height: 36px;">

A aplicação da transformada de Laplace à equação diferencial padrão do sistema mola-massa-amortecedor fornece a saída da seguinte forma:

<img alt="PT_Equation2" src="/files/ftp_upload/16105/16105_PT_Equation_002.svg" style="height: 24px;">

Ao construir o diagrama de blocos, os sinais do lado direito podem ser conectados para simplificar a representação. O diagrama de blocos pode ser refinado ainda mais para incorporar variáveis ​​internas como aceleração e velocidade. Como 1/s corresponde à integração no domínio de Laplace, o sinal de aceleração é integrado para obter a velocidade, e a velocidade é integrada para produzir o sinal de deslocamento.

A simplificação do diagrama de blocos envolve a fatoração de termos do loop de feedback interno. Esse processo leva a um diagrama de blocos alternativo que representa visualmente as relações entre aceleração, velocidade e deslocamento.

Para derivar a função de transferência do sistema, o bloco que representa os sinais de entrada e feedback é movido para o lado direito do comparador, e o loop de feedback interno é simplificado. A equação resultante fornece a função de transferência:

<img alt="PT_Equation3" src="/files/ftp_upload/16105/16105_PT_Equation_003.svg" style="height: 38px;"> 

<img alt="PT_Equation4" src="/files/ftp_upload/16105/16105_PT_Equation_004.svg" style="height: 120px;"> 

Essa função de transferência é essencial para analisar o comportamento do sistema, prever sua resposta a várias entradas e projetar estratégias de controle para atingir as características de desempenho desejadas.

Relação entre equações matemáticas e diagramas de blocos

Considere a equação diferencial de segunda ordem de um sistema mola-massa-amortecedor. O sistema é transformado no domínio de Laplace sob zero inicial ...

relation-between-mathematical-equations-and-block-diagrams

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spring-mass-damper laplace block diagram

A spring-mass-damper system can be written as a second-order differential equation. This equation describes how the system changes over time. When the equation is transformed into the Laplace domain with zero initial conditions, it becomes easier to analyze and rearrange.

The Laplace transform changes a differential equation into an algebraic equation. That makes it simpler to isolate the output. For the standard spring-mass-damper model, the transformed equation gives the output in a form that can be used directly for block diagram construction.

The block diagram is built by connecting the signals on the right-hand side of the equation. It can also include internal variables such as acceleration and velocity. In the Laplace domain, 1/s means integration. So acceleration is integrated to get velocity, and velocity is integrated to get displacement.

The diagram can be simplified by factoring terms from the internal feedback loop. This creates another block diagram that shows the links between acceleration, velocity, and displacement more clearly. To find the transfer function, the block for the input and feedback signals is moved to the right side of the comparator, and the internal feedback loop is reduced. The final transfer function helps describe system behavior, predict the response to inputs, and support control design.

Spring-Mass-Damper Laplace Block Diagram

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A spring-mass-damper system can be written as a second-order differential equation. This equation describes how the system changes over time. When the equation is transformed into the Laplace domain with zero initial conditions, it becomes easier to analyze and rearrange. The Laplace transform changes a differential equation into an algebraic equation. That makes it simpler to isolate the output. For the standard spring-mass-damper model, the transformed equation gives the output in a form...

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Spring-Mass-Damper Laplace Block Diagram

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